- Email: [email protected]
Kinetic models for the simulation of crushing circuits
Mhlerals Enghteerhzg, Vol. 3, No. I/2, pp. 165 180, 1990 Printed in Great Britain
0892-6875/90 $3.00 + 0.00 © 1990 Pergamon Press plc
KINETIC MODELS FOR THE SIMULATION OF CRUSHING CIRCUITS
M. R. MACHADO LEITE Mining Engineering Department - University of Porto, Rua dos Bragas, 4099 Porto Codex, Portugal
ABSTRACT The first part of the paper, 'Fundamentals of Kinetic Models', deals with critical problems concerning modelling: a new parameter condensation technique and phenomenological interpretation; a novel approach for the nonlinearity phenomenon; and a global method for experimental parameter evaluation. As an alternative to the traditional matrix models for crushing simulation, a kinetic type model for Jaw~Cone crushers, in which the residence time is directly related to the crusher's Closed or Open Side Setting, is proposed. For Closed Circuit simulation, instead of the common empirical approaches, a kinetic type model of screening is intended under the novel technique for parameter condensation. Practical conclusions are made from the closed circuit simulator. Keywords Comminution kinetics, screening kinetics, parameter condensation, parameter evaluation, jaw/cone crushers models, screen models. INTRODUCTION We use numerical models of elementary operations and equipments of Minerals Technology not only as predictive tools, but also as gnoseological windows of a complex reality, i.e., as conceptual templates, whose shortcomings in fitting experimental data act as detectors of misadjustments of our theoretical notions. FUNDAMENTALS OF KINETIC MODELS The concept of comminution as a liberation production process instead of as a simple size reduction or surface increasing phenomenon has led to the development of a kinetic theory of comminution and classification [1 ] and has found in Bastenaire's [2] epoch making paper a formal framework allowing a general stochastic approach to comminution in the sense that a multiplicity of unpredictable factors determine every single elementary comminution event. Based on such a formal approach, models have been built for predicting machine and circuit performance, through which automatic control may be implemented, bearing on a wider range of variables and allowing a more accurate manipulation of the overall process performance. 165
166
M . R . MACHADO LE1TE
Kinetic models vs. Matrix models A comminution process, seen as a generator of new particles from pre-existing ones, is adequately described by a general Chapman-Kolmogorov-Smoluchovsky equation C(tz) = T(tl,t2).C(t~)[with te > tl]
[l]
where C(.) represents the particle size distribution at the specified time, and T(.,.) represents the transition function describing how the initial material is broken. The application of function T(.,.) to a given size distribution thus makes for predicting the new particle size distribution. Because such a transition function aims at describing the comminution phenomenon as a whole, it should necessarily include: - the probability of breakage of any particle in the time interval specified; - the probability of appearance of a new particle issued from some pre-existing one, broken in the same time interval. These two probabilities are formally represented by the well known Selection and Breakage functions [K(x) and B(x,x'), respectively ] widely used by all authors. Present comminution models fall into two main types:
Kinetic type, used for grinding machines (tumbling mills, etc.) with strongly varying residence time, in which comminution is considered as a process continuous in time. Thus, Eq. [1] may be rewritten in differential form C(t+dt) = C(t) - F.C(t).dt
[21
where F plays the role of a kinetic constant (for first order kinetics) and is referred to as the instantaneous transition function, which may be obtained from batch tests for different residence times. As may easily be seen, the longer the residence time, the greater will be the extent of transformation of the size distribution. This type of model is commonly and successfully used to adjust experimental data for rod and ball mills, equipment in which there is a strong correlation between feed flow rate and average residence time. Experience with crushers in the coarse size range has led to the notion that these are essentially fixed residence time machines. Although the instantaneous transition may, in this case, still have some theoretical significance, it is not easily obtained from experimental data because batch tests cannot be implemented. So, instead, we use a finite transition, as in Eq. [1], where time is implicit and refers to a single passage of the material through the machine. Experimental determination of such a finite transition can be achieved by means of comminution tests bearing on differently sized feed flows. Models based on this approach are usually referred to as being of the matrix type, because size is discretized into a vector of finite rank, and are commonly used to fit experimental data for jaw and cone crushers [3]. Parameter ¢ondensati0n and phen0menolo~ical interoretation Breaking up the instantaneous Transition matrix into a Selection and a Breakage matrix [{K~} and {Bik}, respectively] makes for a phenomenological description of the coniminution 15rbcess in the sense that it separates the influence of feed material properties from the influence of the mill power configuration. In fact, all our available experience with rod mills [4] tends to demonstrate that the Selection function is closely related with
Simulation of crushing circuits
167
such global features as ore hardness (as measured by Bond's work index, for instance) and mill power (v.g. rod/ball size, drum rotating speed, etc.) while the Breakage function correlates strongly with more subtle features as ore texture and brittleness of grain boundaries. In all practical work, we use Eq. [ 1] or [2] in size-discretized form whereupon the Selection and Breakage functions become matrix operators and size distribution functions become vectors. When using a standard sieve scale of n elements, it will be necessary to determine n elements for the Selection matrix and n . ( n - l ) / 2 elements for the Breakage matrix, thus making for a grand total of n.(n+l)/2 elements. It will be immediately evident that such a formulation, being dependent on the (arbitrary) detail of the size distribution description, is highly irrational. In fact, consideration of the amount of information necessary to describe a first order kinetic process shows that the total number of elements to be independently determined should not be higher than (n-1). Some of the redundancy of the proposed description may, of course, be reduced by considering the normalizing conditions impendent upon size distributions: n of the n.(n1)/2 breakage elements may be derived from the remaining ones by means of the mass conservation equations, thus bringing the grand total down to n . ( n - l ) / 2 , which is still excessive by a factor of ( n - l ) / 2 (which is greater than 1 whenever n is greater than 3). Carefully conducted numerical experiments show that such an excessive number of parametric degrees of freedom in the model brings about: i) difficulties in the convergence of the optimization algorithms used to fit the models to experimental data; ii) instabilities in the solutions, in the sense that small changes in some parameters make for large compensating changes in the others, while the goodness-of-fit and the accuracy of the predictions remain essentially stable; Whereas this is not a big problem from the point of view of circuit design and control, physical interpretation of the obtained parameters becomes utterly unreliable. This, of course, is a highly unsatisfactory situation from the point of view of the experimentalist. In order to overcome these difficulties, we have been working hard on a novel formal technique called parameter condensation which aims at decreasing the number of parametric degrees of freedom by imposing certain physically meaningful constrictions on the matrix elements. These relationships fi(K1, K 2..... Bll, B12..... B22 .... ) = 0 will contain certain parameters (coefficients, exponents), which we call Condensed Parameters and which are, hopefully, less in number than the original matrix elements Ki, Bjk, from them, the latter may be computed. In this way, solution instability may be drastically reduced with the attendant reduction in the amount of experimental data and/or in experimental error. At the present state of the art of mineral processing modelling and simulation, we feel that the development of such a Parameter Condensation technique is a central critical point calling for careful consideration by all researchers in the field. Although our own experience has been a very rich and rewarding one, we still do not feel in possession of a firm grip on the subject. Some partial results may, however, be pointed out:
168
M . R . MACHADOLEITE
the constricting relationships to be imposed to the matrix elements must always be directly supported by experience and by physical interpretation; -
these relationships must conform to the parsimony rule: they must be simple and meaningful; -
- they must also conform to dimensional homogeneity; - condensed parameters should be physically interpretable (i.e., should belong to a phenomenological description of the process) but not directly measurable from experimental kinetics proper. The last recommendation is not yet provable in the strict logical sense, but our experience shows that, in a modelling algorithm, whenever it is not fulfilled the solution to the problem will remain unstable. For example, some measure of the mill power input might make a good condensed parameter but the destruction rate of some size class, however related to mill power, will always make for poor condensation. Approach to the non-linear phenomena The assumption of first order kinetics (the linear approach) means the essential invariability of the transition function along the comminution process. In many experimental cases, first order kinetic models do not provide adequate fitting. In order to obtain better fitting results, some authors have tried empirical kinetic models of orders other than the first [5]. Our approach to this problem has been quite different: we directly assume the transition function to be dependent on time and/or grinding conditions, including the hold-up size distribution itself. This dependence may, for instance, be directly related the degradation of grinding conditions: a)
directly, with the consumption of grinding bodies;
b)
indirectly, with the varying degree of saturation of the grinding loci at any point in the mill, dependent on the feed rate;
c)
indirectly, also, with the interference of some size classes in the breakage of others.
This last approach is a particularly powerful tool to describe the non-linearity of comminution because the evolution of the size composition of the mill is, itself, the essence of grinding. U n d e r this assumption, the transition function will change along comminution time so far as the amount of a definite size will vary, that interferes with the comminution of another size. This type of interference can be observed whenever particles of different sizes are competing for the access to the active grinding zones in the mill or else when definite size classes either hinder or further the destruction of other size classes. The influence of both these phenomena upon the transition function is described as a dependence of the selection function on the size distribution of the mill contents, in conditional form: ~t
Sj IC = Sj .(1 + P~ Kj,m.Cm) where S:* is the mth element of the selection matrix, Cj is the size distribution vector, C m is the weight fraction of size class m in the hold-up (component m of the vector C), and Kj,m expresses the influence of size class m upon the destruction of size class j.
[31
Simulation of crushing circuits
169
Two basic types of interference mechanism have been notionally identified (see also Leite [4]): i) the protection, or umbrella, type, where coarse sizes prevent the ball/rod action upon smaller particles, its effect being proportional to the amount of protecting particles (Ki,m<0, jm); this mechanism is deemed to be particularly important in dry rod milling. Under these assumptions, a non-linear model may be built, the amount of protectors and shock absorbers being computed for each size class and the revised selection matrix obtained by means of Eq. [3]. We believe this to be an entirely new approach to the non-linearity of comminution, where a conceptually plausible and meaningful phenomenology lends support to a definite formulation of the selection matrix, instead of empirically fitting some kinetic order exponent without physical meaning. This type of approach is being particularly useful for building models of the autogenous grinding process, in a theoretical/experimental work undertaken in our laboratory. The only case where a kinetic order other than the first may be acceptable on physical grounds is where saturation of the active grinding zones is suspected or demonstrated, a zero order (i.e., destruction rate independent of the amount of breakable material) model then being in order. Our view to the non-linear character of the general comminution process, however, goes even further than this: we think non-linearity must be its intrinsic fundamental feature because the essential unpredictability of the process strongly suggests that some errorpropagating mechanism must be at work, the non-linear, bifurcation, mechanism being the most plausible candidate [6]. In fact, the fatal markovian character of Eq. [1] and [2] must be challenged in order to accommodate the empirical fact of the broadening of the confidence channel of predictions with comminution time: bifurcating non-linearity must be of the essence, a structural feature, not a mere accident or epiphenomenon. Global method for parameter evaluation From the start, parameter evaluation and scale-up for kinetic simulation have been a problem. In the past, a lot of researchers used to grind narrow size classes alone by themselves in the mill and for very short times, in order to determine the selection and breakage values for each size. As these tests were conducted under conditions rather different from the industrial ones, the technique soon became suspect and a new one, involving radioactivation and tracing of single size classes in a composite feed, was developed in order to allow testing whole, normal, charges in the mill. Because radioactive tracing is a sophisticated and expensive technique available only to well equipped laboratories and practicable only upon certain well-defined materials, some researchers introduced the idea of particle-by-particle tests in a pendulum machine in order to assess the breakage matrix, the selection matrix then being fitted to more conventional experimental data [7].
170
M . R . MACI-IADOLEITE
The method we have been developing in our laboratory, and are now proposing as an alternative, is made up of three main recurring steps: i) the building of a physically consistent initially linear model, supported by good parameter condensation providing a physical meaning for each and every parameter; ii) the global fitting, by means of a powerful and sensitive optimization algorithm, of such a model to superabundant experimental data from batch grinding tests, for widely different residence times, of whole charges ; iii) conceptual control of the results of the fitting routine, extending, where called for, to the introduction of appropriate, sensible non-linear mechanisms. The first step is a very critical and difficult one: as a matter of fact easy physical readability of the condensed parameters is a "sine qua non" condition for the conceptual control of the results (third step). The optimization method we have been using with consistent success for a number of years is the Levenberg-Marquardt algorithm [8] which combines Cauchy and Gauss-Newton methods in variable proportions in order to minimize a sum-of-squares target function. Given consistent models, the use of this routine has been shown to be very successful as far as solution stability (including insensitivity to the initial guess) is concerned; conversely, the routine is highly sensitive to inconsistencies in the model formulation, mainly insensitivity intervals in some parameters, multiple local minima, etc., thus providing a good cross-check for the initial step of the process. Under similar power/charge ratios the kinetic parameters may usually be extrapolated from laboratory to industrial conditions; the scale-up problem then reduces to fitting an adequate transport model to the data from steady state continuous flow tests of industrial machines. Leite [4] presented and demonstrated this method for the global evaluation of rod mill kinetic parameters, including non-linearities. JAW AND CONE CRUSHER KINETIC MODELS Main relevant operational features of iaw and cone crushers A comminution machine may be described as an active volume where mechanical power is input by means of comminuting bodies; the size distribution of the final product will vary with the residence time (or residence time distribution) of particles inside the machine. Residence time is always dependent on the internal transport mechanism provided by the machine design. As far as transport mechanisms, two main classes may be identified [9]: a) jaw and cone crushers are defined by an internal transport not dependent on feed flow rate; in fact, the path of any single particle inside the crusher depends on the jaw/cone geometry and movement, being fairly well-defined and independent of feed flow rate; b) tumbling mills with either overflow or grate discharge have loosely defined paths for both comminuting bodies and mineral particles, dependent mainly on particle and fluid interactions, residence times being thus strongly dependent both on feed flow rate and dilution. As a matter of fact, it is usually assumed that, as opposed to tumbling mills, crushers are machines with a very narrow range of variation for the residence time, corresponding to the also narrow range of settings allowed. Another intrinsic feature of crushers is the size-sensitivity of the internal transport mechanism on account of the decreasing cross-section of the crushing cavity towards the
Simulation of crushing circuits
171
discharge, which hinders the progress of the coarser particles, thus comparatively increasing their probability of being selected for destruction. Because jaw/cone motion is well defined by the mechanical design of the machine, particles are submitted to different types of crushing events, according to their size: a) large particles, easily caught by the jaws/cones, are crushed under high energy levels making for the production of daughter particles of intermediate size; b) lesser particles, able to be caught only between other particles or between other particles and jaw/cone, will be crushed under low energy levels (resulting from energy degradation during propagation) thus making for the production of comparatively larger fines production. Crusher modelling for continuous flow Jaw and cone crushers have usually been described as machines for continuous flow alone and, thus, are never operated under batch conditions; they are also described as fixed residence time machines. On account of these features, many authors [3,10] have felt justified in proposing matrix models for general crusher simulation. Particularly illustrative of this line of thought is the Selective Recirculating Model, as implemented at Mount Isa Mines (Fig. 1) [3], which is built out of i) a size classification step [CI] that selects particles for breakage; ii) another size classification stage [C2] that selects particles for each class of breakage events; iii) and iv) the two breakage events [BI] [B2] described above; v) mass balancing equations. The real usefulness of a simulation model is measured by its ability to accurately predict behaviours outside the experimental range upon which it was built.
tclj [C;
[B1
EB2
Fig.l general matrix model for crushing simulation From the start we think of size segregation inside crushers as a non-linear phenomenon and treat it accordingly. Thus, instead of a matrix model, driven by a finite transition matrix for some fixed, although unspecified residence time, we went for a kinetic model in which the residence time is fitted (not measured) as any other condensed parameter directly related to the crusher's open or closed side setting (O.S.S. or C.S.S). Conceptual control of the modelling will take into consideration the fact that, for instance, the smaller the setting to which the experimental size distribution refers, the longer the residence time that should be fitted.
172
M.R. MACHADO LEITE
As a matter of fact, although crushers do not work under batch conditions, we may always imagine a pseudo-batch work: if the feed flow were suddenly stopped, particles just entering the comminution chamber would spend a definite time before exiting. This time will increase as the closed side setting diminishes. Conceptually, the model is based on an instantaneous transition function and an unspecified average residence time and the prediction of the product size distribution for a given setting (C.S.S.) being obtained through integration of Eq. [2] for a time interval equal to the residence time fitted as a parameter. The model is, thus, built up from a selection, or destruction-rate function that selects particles to be broken under two or three energy levels and the corresponding two or three breakage, or appearance functions. From the point of view of parameter condensation, the two kinetic functions, discretized into kinetic matrixes, may be adequately described as follows: the selection matrix elements, K i , for particles of size i, are given by
-
Ki, j = Ki.I,j.P
j
[4]
where Pi [J = 1, 2, 3 according to the energy level] is a condensed parameter varying in the interval ]0,1[; since Eq. [4] is a recurrent equation, the first element, K 1 1 -- Pa should also be looked upon as another condensed parameter measuring the maximuih energy output of the machine; as will be seen, the other breakage parameters, K I ~ and K1 ~, which have no clear physical meaning, are not independent and consequently-can not'be regarded as condensed parameters; the breakage matrix element, B i k, meaning the weight fraction of daughter particles of size class k issued from a parent iJ~irticle of size class i after the elementary comminution event, is computed from a two parameter H A R R I S size distribution function for each column (i = constant); these two condensed parameters are directly related to the mechanical properties of the rock, since they measure its propensity for producing more or less fine particles. -
These assumptions are partially supported by the well-known Greenwood-Hiorns [11] rule (see also Madureira [12]). The use of a size distribution law for each column in the breakage matrix is also supported by their possible interpretation as the result of comminution of particles of a single size class. Eq. [4] is also lent some support by Griffith's [13] theory of crack initiation and propagation: larger particles are preferentially selected for breakage because they have a high probability of flaw occurrence. The three energetic levels invoked are as follows: i) particles of size greater than the open side setting (O.S.S.) are broken under a high destruction rate, corresponding to a high energy level; ii) particles of size less than the open side setting (O.S.S.) but greater than the closed side setting (C.S.S.) are broken under an intermediate energy level; iii) finally, particles of size less than the closed side setting (C.S.S.) are broken under a very low destruction rate or, even, will skip this stage. As can easily be seen, breakage rate parameters and residence time are not strictly independent, in the sense that both refer to some unit measure of time, so that some kind of restriction should be imposed upon them; in all the work done, we decided to fix the lower residence time and let all other parameters vary freely within their natural domains.
Simulation of crushing circuits
173
Figure 2 represents the selection function as a stepwise defined function for each one of the size (and energy) intervals and shows how it is defined by the four parameters, Pa' P1, PZ' P3"
Pe
__
KI=KI_
1 x
P1
~KI=KI_ j 51 52 ... OSS
1 x P2 I~'~I=K~_I x P3 C55
5n
Fig.2 Selection/Destruction-Rate function parameter condensation Because the final predicted size distribution will be reached by means of an integration over a finite time interval, the word "particle" in the three above items may theoretically mean both an original parent particle or an intermediate parent particle. On account of this we have proposed for this model the designation of selective circulating model in opposition to that implemented in Mount Isa Mines. The great advantages of this model are: the global and consistent evaluation of all parameters; and the explicit reference of each setting by a fitted residence time, allows prediction of machine behaviour for intermediate, not-experimented settings by means of interpolation of the corresponding residence times. Parameter evaluation for fitting models to data We shall now present some simulation results from a model fitted to standard cone crusher data. In order to achieve some simplicity, the model will be driven only by two energetic levels for destruction rate (referred to de O.S.S.) and one breakage function, amounting to a total of three parameters for selection (Pa, P1 and Pa); two parameters for breakage (P ~/ a n d P g ); , . one more parameter (residence time) for each available setting, except for the highest one, which shall be imposed. Some relevant conclusions can be drawn: P_ always fits to the maximum (the unity), meaning that for the coarsest size class tl~e fastest destruction is needed; P1 fits to a high value (ca. 0.9) while P2 fits to a low value (0.3), meaning that for the size classes above the O.S.S. value the destruction rate falls off slowly with increasing mesh number, while the decrement is fast for those below the O.S.S. value; P. and Pg values show that daughter particles fall mainly into the adjacent size J'asses. Figures 3 and 4 show the goodness-of-fit for two different feed size compositions; it can be seen that the larger amount of fines in one of the feed materials does not force any noticeable change in the kinetic parameters, as expected.
ME3-12--L
174
M.R. ~TION
MACHADO LEITE
~TION : ~: 1.i PI: 9.956 ~ - 9,3566 : ~ - 8,929 ~ : 8,256
SELECTIONFUNCTION: Pa: 1.966 PI: 6,939 P'2:-6.3619 OREAKAGEFUNCTION: P~- 6.935 Pg: 6.265
~TION
I 181 90 81 711 m 511
181 81 M 711
//
M 51
//
z'
/111 /
///,,,'
411 31
/ 18
2
I I
¢
, I I I
CSS M • 16 [] I1 • B
CSS i n • 16 r~ 11 • 8
_ _ __ FEED 5II4tLATED
Fixed Time = 10.6 I Fitted time 11.35 j Fitted time 12.11 I
....
RAW DATA
' Fixed Time = 10.6 Fitted Time : 11.47 Fitted Time = 12.37
FEED SIHLLATED RAIl DATA
---....
i
I
I
1
I
I
I
I
I
I
I
I
I
I
I
2
2.1 2,0 4.0 5.6 e.i 11.3 16.0 72,6 ~ , l 4S,2 64.0 M.4 129 161 14 13 12 11 II 9 9 7 6 5 4 3 2 I
I
HH CR
I
I
I
I
I
I
I
I
I
I
I
I
I
2.0 2,9 4.0 5.6 9,1 11,3 16,1 ~,6 3~,0 4S.2 64.0 9D.4 1~9 161 14 13 12 II 11 9 8 7 6 5 4 3 2 I
HH DR
Figs.3 and 4 Condensed parameter evaluation- bulk fitting Figure 5 shows the result of simulated crushing of a feed material containing an abnormal proportion of fines; obviously, the predicted product is much too different from the standard curve; if, however, the above normal proportion of fines were calculated and subtracted from the predicted size composition, the prediction may be seen to be good enough.
Simuleted crushing of an ebnoraely Fine Feed , I t h Condensed Parameters evelueted from e Norael Feed
I 181 91 81 711 66 ra
/ /
__
__ ___
/
/
/
....
_
Feel (Hnrael l Firm) Stendurd Product Curve Product Product without fines
i
i
z
i
z
i
i
I
i
i
i
!
2.1 14
2,9 13
4,1 12
S.6 11
9.0 10
11.3 9
16.6 6
~°6 7
3~.1 C
45.2 5
64.0 4
90.4 )
Fig.5 Accuracy of model prediction
z
129 2
i
161
m
Simulation of crushing circuits
175
V I B R A T I N G S C R E E N K I N E T I C M O D E L F O R C O N T I N U O U S FLOW Predicting crusher circuit performance by a simulation technique needs not only comminution models but also screening models for continuous flow. Research, however, has not paid attention enough to screen modelling, perhaps because screening is reputed to be a simple phenomenon. Lynch & Whiten [3] have developed a model based on the probability that a particle of size s is retained on a screen of aperture h in m trials:
[5]
[1 - ((h-s)/(h+d))2] m
which provides a roughly satisfying partition (Tromp) curve. At Mount Isa Mines, the parameter m has been found to be dependent on the length of the screen and its load. Screenin~ orocess: basis for a ohenomenolo~ical model According to some authors, a continuous flow vibrating screen can be described as having two zones of different kinetics. While stratification is found to play an import part on the fines cleaning of the oversize along the first part of the deck, separation by repetitive trials is the well-known effect that allows the removal of the isosize fraction from the oversize during the residence time of particles in the second part of the deck. These phenomena which were identified in a continuous screen have been analytically described by means of two different processes (see Ferrara [14]): i) the stratification zone has been treated as a saturation process of zero order kinetics; ii) the repetitive trial, or non saturation zone has been treated as a straightforward process of first order kinetics. As a matter of fact, in the stratification zone there are too many particles in competition for the small amount of available openings and, on account of this situation, the flow rate of particles through the screen is at a maximum, independently of the total amount of particles that are attempting to pass through it. Conversely, because in the second part of the deck there are enough openings for the total amount of particles over the screen; the flow rate for any size class may then be assumed proportional to its total amount of particles, the proportionality constant been dependent on the effectiveness of the vibrating action of the screen deck. Figure 6 presents our proposed kinetic model and, according to the legend, d m l ( i ) / d t = -Kf(i)
[zero order kinetics]
[6]
where Kf(i) is the flow rate through the screen for size class i and t is time. On account of the stratification mechanism, Kf(i) will depend on: i) the probability of contact with the screen for a particle of size class i, which is obviously dependent on size i and on the total amount of particles of size equal or less than size i: ContProb(i) = ContProb(i+ 1)[ 1 - k e. ~ j ,= i C(j)]
[7]
where k e is a condensed parameter related to bed stratification efficiency; ii) the probability of passage through the screen, given contact, which depends on the screen aperture according to an expression similar to Eq. [5], in which the number of trials will depend on the residence time according to
176
M.R. MACHADOLEITE
m = k t x frequency of vibration x residence time
[8]
where K t is another condensed parameter acting as a scaling factor related to particle properties such as shape, roughness, etc. iii) the total amount of saturating particles in the first part of the deck, which depends on the feed flow rate: W = (feed flow rate/max feed flow rate) Ks
[9]
where K . is a third condensed parameter which manipulates the shape of the function: its value describes the transport mechanism of particles over the screen induced by the circular, elliptical, linear, etc. motion of its surface.
FEEDIml(i) SATURATION ZONE
u2(1)
m2(i)
NON-SATURATION ZONE
l u3(I)
]
m3(I) t~ OVERSI Z
UNDERSIZE
Fig.6 Vibrating screen kinetic model continuous flow Knowledge of this last parameter allows the calculation of the mean residence time of particles in the stratification zone, a value which is needed for the integration of Eq. [6] in order to obtain the total amount of particles remaining over the screen. These are now able to start separation by repetitive trials under first order kinetics: dm2(i)/dt = _ B(i).m2(i )
[lO]
where B(i) is itself a probability of passage, since contact is now guaranteed. The mean residence time is calculated in the same way as before and integration of Eq. [10] makes for computing the final oversize product. The parameter condensation technique is supported by the physical description of the phenomena at hand. Three dimensionless parameters are commonly used: two for scaling the probabilities of contact (K~) and of passage after contact (kt) and another one for the definition of the total amount b f charge under saturation as a ftinction of feed rate (Ks). Simulation with the model: parameter interpretation For simulating a screen 4 meters long and 1.5 meters wide an algorithm was built where the three parameters were previously fixed, while feed flow rate, feed size distribution and aperture were input values to be manipulated by the user. Table 1 presents some values, including partition curves, from two numerical experiments with all factors constant except feed flow rate. Its analysis leads to the following conclusions: i) the higher the feed rate, the worse the partition obtained;
Simulation of crushing circuits
177
ii) the higher the feed flow rate, the greater the mass fraction of material under saturation (W = 0.51% for Q = 50 tph, against W = 0.72 % for Q = 100 tph); iii) the higher the feed flow rate, the lower the residence times in both zones; iv) partition curves show the amount of misplaced particles in each case; v) columns labelled "% clean 2" and "% clean 3" show the screening efficiency in products m2(i) and mx(i) for each size i; it will be seen that saturation is useful for the oversize fines cleaning (the smaller the size, the higher the efficiency) on account of stratification; separation by repetitive trials is helpful for removing the isosize fraction from the oversize product (the larger the size, the higher the efficiency). TABLE 1 Vibrating screen model- numerical simulation SCREEN: 4 meters long and 1.5 meters wide HOLE SIZE : 8 mm SANE SET OF CONDENSEDPARANETERS TESTS WITH 2 DIFFERENTFEED FLOWRATES Feed = 50 T/h Saturation area = 51% RESIDENCE TINES (calculated): - saturation zone: O.OOBhours - non-saturation zone: O.OOB hours I 1 2 3 4
5 6 7 8 9
mm 22.630 16.000 11.310 8.000 5.660 4.000 2.830 2.000 1.410
CONDENSEDPARANETERS: - Prob. of contact Kc = 0.5 - Prob. passage given contact Kt = 0.05 - Transport mechanism Ks = 0.5
FEED OVERSIZE UNDERSIZE PARTITION 0.130 0.160 0.000 1.000 0.270 0.332 0.000 1.000 0.260 0.320 0.000 1.DO0 0.135 0.166 0.000 1.000 0.065 0.018 0.267 0.230 0.035 0.002 0.177 0.054 0.035 0.001 0.183 0.023 0.010 0.000 0.053 0.004 0.060 0.000 0.320 0.000
Feed = 100 T/h Saturation area = 72 X RESIDENCE TINES (calculated): - saturation zone: 0.006 hours - non-saturation zone: 0.002 hours I 1 2
S 4
5 6 7 8 9
em 22.630 16.000 11.310 8.000 5.660 4.000 2.830 2.000 1.410
ml(I)
0.051 0,106 0,102 0.053 0,025 0.014 0.014 0.004 0,024
m2(I) 0.051 0.106 0.102 0.053 0.012 0.002 0.001 0.000 0.000
0.051 0.i06 0.102 0.053 0.006 0.001 0.000 0.000 0.000
0.00 0.00 0.00 0,00 0.~ 0.86 0.94 0.99 1.00
0.00 0.00 0.00 O.O0 0,24 0.09 0.04 0.01 0.00
CONDENSEDPARAMETERS: - Prob. of contact Kc = 0.5 - Prob. passage given contact KL = 0.05 - Transport mechanism Ks = 0.5
FEED OVERSIZE UNDERSIZE PARTITION 0.130 0,157 0,000 1.OOO 0.270 0.327 0.000 1.000 0.260 0.315 0.000 1.DO0 0.135 0.163 0.000 1.000 0.065 0,033 0.218 0,417 0.035 0.004 0.183 0.091 0.035 0.001 0.1% 0.029 0.010 0.000 0.057 0.004 0.050 0.000 0,345 0.000
el(I) 0.072 0.150 0.144 0.075 0.036 0.019 0.019 0.006
a2(1) 0.072 0.150 0,144 0.075 0.020 0.003 0.001 0,000
0.033
0.000
PARTITION CURVES
1 - Feed Flow Rate = 50 T/h
m3(I) % clean 2X clean3
2 - Feed Flow Rate = 100 T/h
m3(I) Z clean 2Z clean3 0.072 0.00 0.00 0.150 O.O0 0.00 0.144 0.00 O.O0 0.075 0.00 O,O0 0.015 0.44 0.14 0.002 0.82 0.09 0.001 0.93 0.04 0.000 0,99 0.01 0,000 1.00 0.00
178
M, R. MACHADO LEITE
The last conclusion is a very important one that makes for validation of the model as a predictor of screen behaviour. Also, routines for industrial screen selection should include the computational of an additional screen length for work under non-saturation conditions, thus allowing for an effective decrease in the amount of misplaced particles. Since these are mainly isosize, good judgement should be applied as to the effective advantage for downstream processes of such further removal. CLOSED C I R C U I T S I M U L A T I O N In this last section, we present an algorithm for closed circuit simulation as a module for a crushing plant simulator. In our approach, the cone crusher capacity formulae are still missing, but if we get more data from industrial plants, it should be easy to obtain such a module, since our approach involves a residence time measure for each setting, which no other approach does. Another weakness of our model is the absence of a module for the computation of energy input; although this should be easy to overcome, we do not really feel this to be too serious since energy consumption is not the main problem, but rather the production of particles with an adequate size distribution, which parametric and structural circuit design through simulation allows; in fact, excessive fines production is chiefly responsible for both increased energy consumption and bad recovery. Flowsheeting and strategy for the simulation algorithm The flowsheet implemented is shown in Figure 7 and includes the crusher and vibrating screen models, and a procedure for mixing the circulating load with the new feed.
Procedure CRUSHER FEEl)SCREEN
I
Procedure SCREEN J
OVERSIZE I
IUNDRESZZE
Fig.7 Closed circuit flowsheet The solution for each system of linear differential equations is obtained by analytical integration and the various models are linked together forming a steady state configuration by means of a pseudo-dynamic strategy originally used by Madureira [12] where, after each computational cycle, the computed circulating load is added to the fresh feed flow until the final undersize flow rate equals the feed flow rate within a p r e - d e f i n e d precision. Once this point is attained, if feed characteristics were to be kept constant, the size distribution of the final product should also become stable. Discussion of results A variety of simulations have been performed for feed flow rates between 50 and 100 tph and for two screens with different efficiencies.
Simulation of crushing circuits
179
The numerical experiments were performed under the following conditions: - pre-defined parameters for crusher simulation, including the residence time for each setting selected; - two sets of pre-defined parameters for screen simulation, one for very good performance and another one for a poor efficiency;
- feed flow rates chosen by the user within a rage of 50 to 100 tph; - screen aperture freely selected by the user. The main conclusions to be drawn are: - circulating load increases with feed flow rate only on account of the screen inefficiency, since the crusher residence time is independent of feed flow rate up to the choking point; circulating load increases more than proportionally to the feed flow rate because the screen efficiency decreases with increased throughput; the less efficient the screen, the greater the fines production: the recirculation to the crusher of misplaced particles of size slightly above the hole size makes for increased production of particles well bellow the nominal product size, i.e., the hole size. These results clearly show the role played by screen efficiency in crushing circuit performance; this effect is particularly more important in crushing than in grinding, because residence time in the crusher, and thus breakage intensity do not decrease with increased throughput. REFERENCES .
Madureira, C.M.N., et al, Size, Grade and Liberation: A Stochastic Approach to the Fundamental Problem of Mineral Processing. XVI International Mineral Processing Congress, Stockholm. Elsevier Science Publishers, Amsterdam (1988)
.
Bastenaire, F., Etude Th6orique du Mode de G6n6ration des Distributions Granulom6triques. Revue de L'lndustrie Minerale, Janvier, 1965.
.
Lynch, A.J., Mineral Crushing and Grinding Circuits. Elsevier, Amsterdam (1977) Leite, M.R.M., Moagem Nr~o Linear em Moinho de Barras. Ph. D. Thesis, University of Porto, Porto, Portugal (1984)
4.
.
Mika, T.S., Berlioz, L.M., & Fuerstenau, D.W., An Approach to the Kinetics of Dry Batch Ball Milling. 2nd European Symposium of Size Reduction. Verlag Chemie, Weinheim (1967)
.
Rocha E. Silva, J.A.C., Calibre: Conceito e Medida. Ph. D. Thesis, University of Porto, Porto, Portugal (1989)
.
Lynch, A.J., Simulation - The Design Tool for the Future. In Mineral Processing at a Crossroads (ed. B.A. Wills & R.W. Barley), Martinus Nijhoff, Dordrecht (1986)
.
Nash, J.C., Compact Numerical Methods for Computers. Adam Hilger, Ltd, Bristol (1979)
.
Leite, M.R.M, Circuitos de Fragmenta~o Grauda. Boletim de Minas, 21/2, Lisboa (1984)
180
M. R. MACHADO LEITE
10.
Whiten, W.J., The Simulation of Crushing Plants with Models Developed using Multiple Spline Regression. Journal of the South African Institute of Mining and Metallurgy, (May 1972)
11.
Greenwood, J., & Hiorns, F.J., A Comparison of Individual and Collective Breakage of Particle Assemblies. 2nd European Symposium of Size Reduction, Verlag Chemie, Weinheim (1967)
12.
Madureira, C.M.N., et al, Conceitos Fundamantais para o Estudo Fenomenol6gico dos Processos de Fragmenta~o. Modelos de Simula~o. II Simp. Sobre Teorias da Informa~o e dos Sistemas, Faculdade de Engenharia, University of Porto, Portugal (1972)
13.
Griffith, A.A., The Phenomena of Rupture and Flow in Solids. Phil Trans. A. (1921)
14.
Ferrara, G. et al, A Contribution to Screening Kinetics. XI International Mineral Processing Congress, University of Cagliari, Cagliari (1975)
0892-6875/90 $3.00 + 0.00 © 1990 Pergamon Press plc
KINETIC MODELS FOR THE SIMULATION OF CRUSHING CIRCUITS
M. R. MACHADO LEITE Mining Engineering Department - University of Porto, Rua dos Bragas, 4099 Porto Codex, Portugal
ABSTRACT The first part of the paper, 'Fundamentals of Kinetic Models', deals with critical problems concerning modelling: a new parameter condensation technique and phenomenological interpretation; a novel approach for the nonlinearity phenomenon; and a global method for experimental parameter evaluation. As an alternative to the traditional matrix models for crushing simulation, a kinetic type model for Jaw~Cone crushers, in which the residence time is directly related to the crusher's Closed or Open Side Setting, is proposed. For Closed Circuit simulation, instead of the common empirical approaches, a kinetic type model of screening is intended under the novel technique for parameter condensation. Practical conclusions are made from the closed circuit simulator. Keywords Comminution kinetics, screening kinetics, parameter condensation, parameter evaluation, jaw/cone crushers models, screen models. INTRODUCTION We use numerical models of elementary operations and equipments of Minerals Technology not only as predictive tools, but also as gnoseological windows of a complex reality, i.e., as conceptual templates, whose shortcomings in fitting experimental data act as detectors of misadjustments of our theoretical notions. FUNDAMENTALS OF KINETIC MODELS The concept of comminution as a liberation production process instead of as a simple size reduction or surface increasing phenomenon has led to the development of a kinetic theory of comminution and classification [1 ] and has found in Bastenaire's [2] epoch making paper a formal framework allowing a general stochastic approach to comminution in the sense that a multiplicity of unpredictable factors determine every single elementary comminution event. Based on such a formal approach, models have been built for predicting machine and circuit performance, through which automatic control may be implemented, bearing on a wider range of variables and allowing a more accurate manipulation of the overall process performance. 165
166
M . R . MACHADO LE1TE
Kinetic models vs. Matrix models A comminution process, seen as a generator of new particles from pre-existing ones, is adequately described by a general Chapman-Kolmogorov-Smoluchovsky equation C(tz) = T(tl,t2).C(t~)[with te > tl]
[l]
where C(.) represents the particle size distribution at the specified time, and T(.,.) represents the transition function describing how the initial material is broken. The application of function T(.,.) to a given size distribution thus makes for predicting the new particle size distribution. Because such a transition function aims at describing the comminution phenomenon as a whole, it should necessarily include: - the probability of breakage of any particle in the time interval specified; - the probability of appearance of a new particle issued from some pre-existing one, broken in the same time interval. These two probabilities are formally represented by the well known Selection and Breakage functions [K(x) and B(x,x'), respectively ] widely used by all authors. Present comminution models fall into two main types:
Kinetic type, used for grinding machines (tumbling mills, etc.) with strongly varying residence time, in which comminution is considered as a process continuous in time. Thus, Eq. [1] may be rewritten in differential form C(t+dt) = C(t) - F.C(t).dt
[21
where F plays the role of a kinetic constant (for first order kinetics) and is referred to as the instantaneous transition function, which may be obtained from batch tests for different residence times. As may easily be seen, the longer the residence time, the greater will be the extent of transformation of the size distribution. This type of model is commonly and successfully used to adjust experimental data for rod and ball mills, equipment in which there is a strong correlation between feed flow rate and average residence time. Experience with crushers in the coarse size range has led to the notion that these are essentially fixed residence time machines. Although the instantaneous transition may, in this case, still have some theoretical significance, it is not easily obtained from experimental data because batch tests cannot be implemented. So, instead, we use a finite transition, as in Eq. [1], where time is implicit and refers to a single passage of the material through the machine. Experimental determination of such a finite transition can be achieved by means of comminution tests bearing on differently sized feed flows. Models based on this approach are usually referred to as being of the matrix type, because size is discretized into a vector of finite rank, and are commonly used to fit experimental data for jaw and cone crushers [3]. Parameter ¢ondensati0n and phen0menolo~ical interoretation Breaking up the instantaneous Transition matrix into a Selection and a Breakage matrix [{K~} and {Bik}, respectively] makes for a phenomenological description of the coniminution 15rbcess in the sense that it separates the influence of feed material properties from the influence of the mill power configuration. In fact, all our available experience with rod mills [4] tends to demonstrate that the Selection function is closely related with
Simulation of crushing circuits
167
such global features as ore hardness (as measured by Bond's work index, for instance) and mill power (v.g. rod/ball size, drum rotating speed, etc.) while the Breakage function correlates strongly with more subtle features as ore texture and brittleness of grain boundaries. In all practical work, we use Eq. [ 1] or [2] in size-discretized form whereupon the Selection and Breakage functions become matrix operators and size distribution functions become vectors. When using a standard sieve scale of n elements, it will be necessary to determine n elements for the Selection matrix and n . ( n - l ) / 2 elements for the Breakage matrix, thus making for a grand total of n.(n+l)/2 elements. It will be immediately evident that such a formulation, being dependent on the (arbitrary) detail of the size distribution description, is highly irrational. In fact, consideration of the amount of information necessary to describe a first order kinetic process shows that the total number of elements to be independently determined should not be higher than (n-1). Some of the redundancy of the proposed description may, of course, be reduced by considering the normalizing conditions impendent upon size distributions: n of the n.(n1)/2 breakage elements may be derived from the remaining ones by means of the mass conservation equations, thus bringing the grand total down to n . ( n - l ) / 2 , which is still excessive by a factor of ( n - l ) / 2 (which is greater than 1 whenever n is greater than 3). Carefully conducted numerical experiments show that such an excessive number of parametric degrees of freedom in the model brings about: i) difficulties in the convergence of the optimization algorithms used to fit the models to experimental data; ii) instabilities in the solutions, in the sense that small changes in some parameters make for large compensating changes in the others, while the goodness-of-fit and the accuracy of the predictions remain essentially stable; Whereas this is not a big problem from the point of view of circuit design and control, physical interpretation of the obtained parameters becomes utterly unreliable. This, of course, is a highly unsatisfactory situation from the point of view of the experimentalist. In order to overcome these difficulties, we have been working hard on a novel formal technique called parameter condensation which aims at decreasing the number of parametric degrees of freedom by imposing certain physically meaningful constrictions on the matrix elements. These relationships fi(K1, K 2..... Bll, B12..... B22 .... ) = 0 will contain certain parameters (coefficients, exponents), which we call Condensed Parameters and which are, hopefully, less in number than the original matrix elements Ki, Bjk, from them, the latter may be computed. In this way, solution instability may be drastically reduced with the attendant reduction in the amount of experimental data and/or in experimental error. At the present state of the art of mineral processing modelling and simulation, we feel that the development of such a Parameter Condensation technique is a central critical point calling for careful consideration by all researchers in the field. Although our own experience has been a very rich and rewarding one, we still do not feel in possession of a firm grip on the subject. Some partial results may, however, be pointed out:
168
M . R . MACHADOLEITE
the constricting relationships to be imposed to the matrix elements must always be directly supported by experience and by physical interpretation; -
these relationships must conform to the parsimony rule: they must be simple and meaningful; -
- they must also conform to dimensional homogeneity; - condensed parameters should be physically interpretable (i.e., should belong to a phenomenological description of the process) but not directly measurable from experimental kinetics proper. The last recommendation is not yet provable in the strict logical sense, but our experience shows that, in a modelling algorithm, whenever it is not fulfilled the solution to the problem will remain unstable. For example, some measure of the mill power input might make a good condensed parameter but the destruction rate of some size class, however related to mill power, will always make for poor condensation. Approach to the non-linear phenomena The assumption of first order kinetics (the linear approach) means the essential invariability of the transition function along the comminution process. In many experimental cases, first order kinetic models do not provide adequate fitting. In order to obtain better fitting results, some authors have tried empirical kinetic models of orders other than the first [5]. Our approach to this problem has been quite different: we directly assume the transition function to be dependent on time and/or grinding conditions, including the hold-up size distribution itself. This dependence may, for instance, be directly related the degradation of grinding conditions: a)
directly, with the consumption of grinding bodies;
b)
indirectly, with the varying degree of saturation of the grinding loci at any point in the mill, dependent on the feed rate;
c)
indirectly, also, with the interference of some size classes in the breakage of others.
This last approach is a particularly powerful tool to describe the non-linearity of comminution because the evolution of the size composition of the mill is, itself, the essence of grinding. U n d e r this assumption, the transition function will change along comminution time so far as the amount of a definite size will vary, that interferes with the comminution of another size. This type of interference can be observed whenever particles of different sizes are competing for the access to the active grinding zones in the mill or else when definite size classes either hinder or further the destruction of other size classes. The influence of both these phenomena upon the transition function is described as a dependence of the selection function on the size distribution of the mill contents, in conditional form: ~t
Sj IC = Sj .(1 + P~ Kj,m.Cm) where S:* is the mth element of the selection matrix, Cj is the size distribution vector, C m is the weight fraction of size class m in the hold-up (component m of the vector C), and Kj,m expresses the influence of size class m upon the destruction of size class j.
[31
Simulation of crushing circuits
169
Two basic types of interference mechanism have been notionally identified (see also Leite [4]): i) the protection, or umbrella, type, where coarse sizes prevent the ball/rod action upon smaller particles, its effect being proportional to the amount of protecting particles (Ki,m<0, j
170
M . R . MACI-IADOLEITE
The method we have been developing in our laboratory, and are now proposing as an alternative, is made up of three main recurring steps: i) the building of a physically consistent initially linear model, supported by good parameter condensation providing a physical meaning for each and every parameter; ii) the global fitting, by means of a powerful and sensitive optimization algorithm, of such a model to superabundant experimental data from batch grinding tests, for widely different residence times, of whole charges ; iii) conceptual control of the results of the fitting routine, extending, where called for, to the introduction of appropriate, sensible non-linear mechanisms. The first step is a very critical and difficult one: as a matter of fact easy physical readability of the condensed parameters is a "sine qua non" condition for the conceptual control of the results (third step). The optimization method we have been using with consistent success for a number of years is the Levenberg-Marquardt algorithm [8] which combines Cauchy and Gauss-Newton methods in variable proportions in order to minimize a sum-of-squares target function. Given consistent models, the use of this routine has been shown to be very successful as far as solution stability (including insensitivity to the initial guess) is concerned; conversely, the routine is highly sensitive to inconsistencies in the model formulation, mainly insensitivity intervals in some parameters, multiple local minima, etc., thus providing a good cross-check for the initial step of the process. Under similar power/charge ratios the kinetic parameters may usually be extrapolated from laboratory to industrial conditions; the scale-up problem then reduces to fitting an adequate transport model to the data from steady state continuous flow tests of industrial machines. Leite [4] presented and demonstrated this method for the global evaluation of rod mill kinetic parameters, including non-linearities. JAW AND CONE CRUSHER KINETIC MODELS Main relevant operational features of iaw and cone crushers A comminution machine may be described as an active volume where mechanical power is input by means of comminuting bodies; the size distribution of the final product will vary with the residence time (or residence time distribution) of particles inside the machine. Residence time is always dependent on the internal transport mechanism provided by the machine design. As far as transport mechanisms, two main classes may be identified [9]: a) jaw and cone crushers are defined by an internal transport not dependent on feed flow rate; in fact, the path of any single particle inside the crusher depends on the jaw/cone geometry and movement, being fairly well-defined and independent of feed flow rate; b) tumbling mills with either overflow or grate discharge have loosely defined paths for both comminuting bodies and mineral particles, dependent mainly on particle and fluid interactions, residence times being thus strongly dependent both on feed flow rate and dilution. As a matter of fact, it is usually assumed that, as opposed to tumbling mills, crushers are machines with a very narrow range of variation for the residence time, corresponding to the also narrow range of settings allowed. Another intrinsic feature of crushers is the size-sensitivity of the internal transport mechanism on account of the decreasing cross-section of the crushing cavity towards the
Simulation of crushing circuits
171
discharge, which hinders the progress of the coarser particles, thus comparatively increasing their probability of being selected for destruction. Because jaw/cone motion is well defined by the mechanical design of the machine, particles are submitted to different types of crushing events, according to their size: a) large particles, easily caught by the jaws/cones, are crushed under high energy levels making for the production of daughter particles of intermediate size; b) lesser particles, able to be caught only between other particles or between other particles and jaw/cone, will be crushed under low energy levels (resulting from energy degradation during propagation) thus making for the production of comparatively larger fines production. Crusher modelling for continuous flow Jaw and cone crushers have usually been described as machines for continuous flow alone and, thus, are never operated under batch conditions; they are also described as fixed residence time machines. On account of these features, many authors [3,10] have felt justified in proposing matrix models for general crusher simulation. Particularly illustrative of this line of thought is the Selective Recirculating Model, as implemented at Mount Isa Mines (Fig. 1) [3], which is built out of i) a size classification step [CI] that selects particles for breakage; ii) another size classification stage [C2] that selects particles for each class of breakage events; iii) and iv) the two breakage events [BI] [B2] described above; v) mass balancing equations. The real usefulness of a simulation model is measured by its ability to accurately predict behaviours outside the experimental range upon which it was built.
tclj [C;
[B1
EB2
Fig.l general matrix model for crushing simulation From the start we think of size segregation inside crushers as a non-linear phenomenon and treat it accordingly. Thus, instead of a matrix model, driven by a finite transition matrix for some fixed, although unspecified residence time, we went for a kinetic model in which the residence time is fitted (not measured) as any other condensed parameter directly related to the crusher's open or closed side setting (O.S.S. or C.S.S). Conceptual control of the modelling will take into consideration the fact that, for instance, the smaller the setting to which the experimental size distribution refers, the longer the residence time that should be fitted.
172
M.R. MACHADO LEITE
As a matter of fact, although crushers do not work under batch conditions, we may always imagine a pseudo-batch work: if the feed flow were suddenly stopped, particles just entering the comminution chamber would spend a definite time before exiting. This time will increase as the closed side setting diminishes. Conceptually, the model is based on an instantaneous transition function and an unspecified average residence time and the prediction of the product size distribution for a given setting (C.S.S.) being obtained through integration of Eq. [2] for a time interval equal to the residence time fitted as a parameter. The model is, thus, built up from a selection, or destruction-rate function that selects particles to be broken under two or three energy levels and the corresponding two or three breakage, or appearance functions. From the point of view of parameter condensation, the two kinetic functions, discretized into kinetic matrixes, may be adequately described as follows: the selection matrix elements, K i , for particles of size i, are given by
-
Ki, j = Ki.I,j.P
j
[4]
where Pi [J = 1, 2, 3 according to the energy level] is a condensed parameter varying in the interval ]0,1[; since Eq. [4] is a recurrent equation, the first element, K 1 1 -- Pa should also be looked upon as another condensed parameter measuring the maximuih energy output of the machine; as will be seen, the other breakage parameters, K I ~ and K1 ~, which have no clear physical meaning, are not independent and consequently-can not'be regarded as condensed parameters; the breakage matrix element, B i k, meaning the weight fraction of daughter particles of size class k issued from a parent iJ~irticle of size class i after the elementary comminution event, is computed from a two parameter H A R R I S size distribution function for each column (i = constant); these two condensed parameters are directly related to the mechanical properties of the rock, since they measure its propensity for producing more or less fine particles. -
These assumptions are partially supported by the well-known Greenwood-Hiorns [11] rule (see also Madureira [12]). The use of a size distribution law for each column in the breakage matrix is also supported by their possible interpretation as the result of comminution of particles of a single size class. Eq. [4] is also lent some support by Griffith's [13] theory of crack initiation and propagation: larger particles are preferentially selected for breakage because they have a high probability of flaw occurrence. The three energetic levels invoked are as follows: i) particles of size greater than the open side setting (O.S.S.) are broken under a high destruction rate, corresponding to a high energy level; ii) particles of size less than the open side setting (O.S.S.) but greater than the closed side setting (C.S.S.) are broken under an intermediate energy level; iii) finally, particles of size less than the closed side setting (C.S.S.) are broken under a very low destruction rate or, even, will skip this stage. As can easily be seen, breakage rate parameters and residence time are not strictly independent, in the sense that both refer to some unit measure of time, so that some kind of restriction should be imposed upon them; in all the work done, we decided to fix the lower residence time and let all other parameters vary freely within their natural domains.
Simulation of crushing circuits
173
Figure 2 represents the selection function as a stepwise defined function for each one of the size (and energy) intervals and shows how it is defined by the four parameters, Pa' P1, PZ' P3"
Pe
__
KI=KI_
1 x
P1
~KI=KI_ j 51 52 ... OSS
1 x P2 I~'~I=K~_I x P3 C55
5n
Fig.2 Selection/Destruction-Rate function parameter condensation Because the final predicted size distribution will be reached by means of an integration over a finite time interval, the word "particle" in the three above items may theoretically mean both an original parent particle or an intermediate parent particle. On account of this we have proposed for this model the designation of selective circulating model in opposition to that implemented in Mount Isa Mines. The great advantages of this model are: the global and consistent evaluation of all parameters; and the explicit reference of each setting by a fitted residence time, allows prediction of machine behaviour for intermediate, not-experimented settings by means of interpolation of the corresponding residence times. Parameter evaluation for fitting models to data We shall now present some simulation results from a model fitted to standard cone crusher data. In order to achieve some simplicity, the model will be driven only by two energetic levels for destruction rate (referred to de O.S.S.) and one breakage function, amounting to a total of three parameters for selection (Pa, P1 and Pa); two parameters for breakage (P ~/ a n d P g ); , . one more parameter (residence time) for each available setting, except for the highest one, which shall be imposed. Some relevant conclusions can be drawn: P_ always fits to the maximum (the unity), meaning that for the coarsest size class tl~e fastest destruction is needed; P1 fits to a high value (ca. 0.9) while P2 fits to a low value (0.3), meaning that for the size classes above the O.S.S. value the destruction rate falls off slowly with increasing mesh number, while the decrement is fast for those below the O.S.S. value; P. and Pg values show that daughter particles fall mainly into the adjacent size J'asses. Figures 3 and 4 show the goodness-of-fit for two different feed size compositions; it can be seen that the larger amount of fines in one of the feed materials does not force any noticeable change in the kinetic parameters, as expected.
ME3-12--L
174
M.R. ~TION
MACHADO LEITE
~TION : ~: 1.i PI: 9.956 ~ - 9,3566 : ~ - 8,929 ~ : 8,256
SELECTIONFUNCTION: Pa: 1.966 PI: 6,939 P'2:-6.3619 OREAKAGEFUNCTION: P~- 6.935 Pg: 6.265
~TION
I 181 90 81 711 m 511
181 81 M 711
//
M 51
//
z'
/111 /
///,,,'
411 31
/ 18
2
I I
¢
, I I I
CSS M • 16 [] I1 • B
CSS i n • 16 r~ 11 • 8
_ _ __ FEED 5II4tLATED
Fixed Time = 10.6 I Fitted time 11.35 j Fitted time 12.11 I
....
RAW DATA
' Fixed Time = 10.6 Fitted Time : 11.47 Fitted Time = 12.37
FEED SIHLLATED RAIl DATA
---....
i
I
I
1
I
I
I
I
I
I
I
I
I
I
I
2
2.1 2,0 4.0 5.6 e.i 11.3 16.0 72,6 ~ , l 4S,2 64.0 M.4 129 161 14 13 12 11 II 9 9 7 6 5 4 3 2 I
I
HH CR
I
I
I
I
I
I
I
I
I
I
I
I
I
2.0 2,9 4.0 5.6 9,1 11,3 16,1 ~,6 3~,0 4S.2 64.0 9D.4 1~9 161 14 13 12 II 11 9 8 7 6 5 4 3 2 I
HH DR
Figs.3 and 4 Condensed parameter evaluation- bulk fitting Figure 5 shows the result of simulated crushing of a feed material containing an abnormal proportion of fines; obviously, the predicted product is much too different from the standard curve; if, however, the above normal proportion of fines were calculated and subtracted from the predicted size composition, the prediction may be seen to be good enough.
Simuleted crushing of an ebnoraely Fine Feed , I t h Condensed Parameters evelueted from e Norael Feed
I 181 91 81 711 66 ra
/ /
__
__ ___
/
/
/
....
_
Feel (Hnrael l Firm) Stendurd Product Curve Product Product without fines
i
i
z
i
z
i
i
I
i
i
i
!
2.1 14
2,9 13
4,1 12
S.6 11
9.0 10
11.3 9
16.6 6
~°6 7
3~.1 C
45.2 5
64.0 4
90.4 )
Fig.5 Accuracy of model prediction
z
129 2
i
161
m
Simulation of crushing circuits
175
V I B R A T I N G S C R E E N K I N E T I C M O D E L F O R C O N T I N U O U S FLOW Predicting crusher circuit performance by a simulation technique needs not only comminution models but also screening models for continuous flow. Research, however, has not paid attention enough to screen modelling, perhaps because screening is reputed to be a simple phenomenon. Lynch & Whiten [3] have developed a model based on the probability that a particle of size s is retained on a screen of aperture h in m trials:
[5]
[1 - ((h-s)/(h+d))2] m
which provides a roughly satisfying partition (Tromp) curve. At Mount Isa Mines, the parameter m has been found to be dependent on the length of the screen and its load. Screenin~ orocess: basis for a ohenomenolo~ical model According to some authors, a continuous flow vibrating screen can be described as having two zones of different kinetics. While stratification is found to play an import part on the fines cleaning of the oversize along the first part of the deck, separation by repetitive trials is the well-known effect that allows the removal of the isosize fraction from the oversize during the residence time of particles in the second part of the deck. These phenomena which were identified in a continuous screen have been analytically described by means of two different processes (see Ferrara [14]): i) the stratification zone has been treated as a saturation process of zero order kinetics; ii) the repetitive trial, or non saturation zone has been treated as a straightforward process of first order kinetics. As a matter of fact, in the stratification zone there are too many particles in competition for the small amount of available openings and, on account of this situation, the flow rate of particles through the screen is at a maximum, independently of the total amount of particles that are attempting to pass through it. Conversely, because in the second part of the deck there are enough openings for the total amount of particles over the screen; the flow rate for any size class may then be assumed proportional to its total amount of particles, the proportionality constant been dependent on the effectiveness of the vibrating action of the screen deck. Figure 6 presents our proposed kinetic model and, according to the legend, d m l ( i ) / d t = -Kf(i)
[zero order kinetics]
[6]
where Kf(i) is the flow rate through the screen for size class i and t is time. On account of the stratification mechanism, Kf(i) will depend on: i) the probability of contact with the screen for a particle of size class i, which is obviously dependent on size i and on the total amount of particles of size equal or less than size i: ContProb(i) = ContProb(i+ 1)[ 1 - k e. ~ j ,= i C(j)]
[7]
where k e is a condensed parameter related to bed stratification efficiency; ii) the probability of passage through the screen, given contact, which depends on the screen aperture according to an expression similar to Eq. [5], in which the number of trials will depend on the residence time according to
176
M.R. MACHADOLEITE
m = k t x frequency of vibration x residence time
[8]
where K t is another condensed parameter acting as a scaling factor related to particle properties such as shape, roughness, etc. iii) the total amount of saturating particles in the first part of the deck, which depends on the feed flow rate: W = (feed flow rate/max feed flow rate) Ks
[9]
where K . is a third condensed parameter which manipulates the shape of the function: its value describes the transport mechanism of particles over the screen induced by the circular, elliptical, linear, etc. motion of its surface.
FEEDIml(i) SATURATION ZONE
u2(1)
m2(i)
NON-SATURATION ZONE
l u3(I)
]
m3(I) t~ OVERSI Z
UNDERSIZE
Fig.6 Vibrating screen kinetic model continuous flow Knowledge of this last parameter allows the calculation of the mean residence time of particles in the stratification zone, a value which is needed for the integration of Eq. [6] in order to obtain the total amount of particles remaining over the screen. These are now able to start separation by repetitive trials under first order kinetics: dm2(i)/dt = _ B(i).m2(i )
[lO]
where B(i) is itself a probability of passage, since contact is now guaranteed. The mean residence time is calculated in the same way as before and integration of Eq. [10] makes for computing the final oversize product. The parameter condensation technique is supported by the physical description of the phenomena at hand. Three dimensionless parameters are commonly used: two for scaling the probabilities of contact (K~) and of passage after contact (kt) and another one for the definition of the total amount b f charge under saturation as a ftinction of feed rate (Ks). Simulation with the model: parameter interpretation For simulating a screen 4 meters long and 1.5 meters wide an algorithm was built where the three parameters were previously fixed, while feed flow rate, feed size distribution and aperture were input values to be manipulated by the user. Table 1 presents some values, including partition curves, from two numerical experiments with all factors constant except feed flow rate. Its analysis leads to the following conclusions: i) the higher the feed rate, the worse the partition obtained;
Simulation of crushing circuits
177
ii) the higher the feed flow rate, the greater the mass fraction of material under saturation (W = 0.51% for Q = 50 tph, against W = 0.72 % for Q = 100 tph); iii) the higher the feed flow rate, the lower the residence times in both zones; iv) partition curves show the amount of misplaced particles in each case; v) columns labelled "% clean 2" and "% clean 3" show the screening efficiency in products m2(i) and mx(i) for each size i; it will be seen that saturation is useful for the oversize fines cleaning (the smaller the size, the higher the efficiency) on account of stratification; separation by repetitive trials is helpful for removing the isosize fraction from the oversize product (the larger the size, the higher the efficiency). TABLE 1 Vibrating screen model- numerical simulation SCREEN: 4 meters long and 1.5 meters wide HOLE SIZE : 8 mm SANE SET OF CONDENSEDPARANETERS TESTS WITH 2 DIFFERENTFEED FLOWRATES Feed = 50 T/h Saturation area = 51% RESIDENCE TINES (calculated): - saturation zone: O.OOBhours - non-saturation zone: O.OOB hours I 1 2 3 4
5 6 7 8 9
mm 22.630 16.000 11.310 8.000 5.660 4.000 2.830 2.000 1.410
CONDENSEDPARANETERS: - Prob. of contact Kc = 0.5 - Prob. passage given contact Kt = 0.05 - Transport mechanism Ks = 0.5
FEED OVERSIZE UNDERSIZE PARTITION 0.130 0.160 0.000 1.000 0.270 0.332 0.000 1.000 0.260 0.320 0.000 1.DO0 0.135 0.166 0.000 1.000 0.065 0.018 0.267 0.230 0.035 0.002 0.177 0.054 0.035 0.001 0.183 0.023 0.010 0.000 0.053 0.004 0.060 0.000 0.320 0.000
Feed = 100 T/h Saturation area = 72 X RESIDENCE TINES (calculated): - saturation zone: 0.006 hours - non-saturation zone: 0.002 hours I 1 2
S 4
5 6 7 8 9
em 22.630 16.000 11.310 8.000 5.660 4.000 2.830 2.000 1.410
ml(I)
0.051 0,106 0,102 0.053 0,025 0.014 0.014 0.004 0,024
m2(I) 0.051 0.106 0.102 0.053 0.012 0.002 0.001 0.000 0.000
0.051 0.i06 0.102 0.053 0.006 0.001 0.000 0.000 0.000
0.00 0.00 0.00 0,00 0.~ 0.86 0.94 0.99 1.00
0.00 0.00 0.00 O.O0 0,24 0.09 0.04 0.01 0.00
CONDENSEDPARAMETERS: - Prob. of contact Kc = 0.5 - Prob. passage given contact KL = 0.05 - Transport mechanism Ks = 0.5
FEED OVERSIZE UNDERSIZE PARTITION 0.130 0,157 0,000 1.OOO 0.270 0.327 0.000 1.000 0.260 0.315 0.000 1.DO0 0.135 0.163 0.000 1.000 0.065 0,033 0.218 0,417 0.035 0.004 0.183 0.091 0.035 0.001 0.1% 0.029 0.010 0.000 0.057 0.004 0.050 0.000 0,345 0.000
el(I) 0.072 0.150 0.144 0.075 0.036 0.019 0.019 0.006
a2(1) 0.072 0.150 0,144 0.075 0.020 0.003 0.001 0,000
0.033
0.000
PARTITION CURVES
1 - Feed Flow Rate = 50 T/h
m3(I) % clean 2X clean3
2 - Feed Flow Rate = 100 T/h
m3(I) Z clean 2Z clean3 0.072 0.00 0.00 0.150 O.O0 0.00 0.144 0.00 O.O0 0.075 0.00 O,O0 0.015 0.44 0.14 0.002 0.82 0.09 0.001 0.93 0.04 0.000 0,99 0.01 0,000 1.00 0.00
178
M, R. MACHADO LEITE
The last conclusion is a very important one that makes for validation of the model as a predictor of screen behaviour. Also, routines for industrial screen selection should include the computational of an additional screen length for work under non-saturation conditions, thus allowing for an effective decrease in the amount of misplaced particles. Since these are mainly isosize, good judgement should be applied as to the effective advantage for downstream processes of such further removal. CLOSED C I R C U I T S I M U L A T I O N In this last section, we present an algorithm for closed circuit simulation as a module for a crushing plant simulator. In our approach, the cone crusher capacity formulae are still missing, but if we get more data from industrial plants, it should be easy to obtain such a module, since our approach involves a residence time measure for each setting, which no other approach does. Another weakness of our model is the absence of a module for the computation of energy input; although this should be easy to overcome, we do not really feel this to be too serious since energy consumption is not the main problem, but rather the production of particles with an adequate size distribution, which parametric and structural circuit design through simulation allows; in fact, excessive fines production is chiefly responsible for both increased energy consumption and bad recovery. Flowsheeting and strategy for the simulation algorithm The flowsheet implemented is shown in Figure 7 and includes the crusher and vibrating screen models, and a procedure for mixing the circulating load with the new feed.
Procedure CRUSHER FEEl)SCREEN
I
Procedure SCREEN J
OVERSIZE I
IUNDRESZZE
Fig.7 Closed circuit flowsheet The solution for each system of linear differential equations is obtained by analytical integration and the various models are linked together forming a steady state configuration by means of a pseudo-dynamic strategy originally used by Madureira [12] where, after each computational cycle, the computed circulating load is added to the fresh feed flow until the final undersize flow rate equals the feed flow rate within a p r e - d e f i n e d precision. Once this point is attained, if feed characteristics were to be kept constant, the size distribution of the final product should also become stable. Discussion of results A variety of simulations have been performed for feed flow rates between 50 and 100 tph and for two screens with different efficiencies.
Simulation of crushing circuits
179
The numerical experiments were performed under the following conditions: - pre-defined parameters for crusher simulation, including the residence time for each setting selected; - two sets of pre-defined parameters for screen simulation, one for very good performance and another one for a poor efficiency;
- feed flow rates chosen by the user within a rage of 50 to 100 tph; - screen aperture freely selected by the user. The main conclusions to be drawn are: - circulating load increases with feed flow rate only on account of the screen inefficiency, since the crusher residence time is independent of feed flow rate up to the choking point; circulating load increases more than proportionally to the feed flow rate because the screen efficiency decreases with increased throughput; the less efficient the screen, the greater the fines production: the recirculation to the crusher of misplaced particles of size slightly above the hole size makes for increased production of particles well bellow the nominal product size, i.e., the hole size. These results clearly show the role played by screen efficiency in crushing circuit performance; this effect is particularly more important in crushing than in grinding, because residence time in the crusher, and thus breakage intensity do not decrease with increased throughput. REFERENCES .
Madureira, C.M.N., et al, Size, Grade and Liberation: A Stochastic Approach to the Fundamental Problem of Mineral Processing. XVI International Mineral Processing Congress, Stockholm. Elsevier Science Publishers, Amsterdam (1988)
.
Bastenaire, F., Etude Th6orique du Mode de G6n6ration des Distributions Granulom6triques. Revue de L'lndustrie Minerale, Janvier, 1965.
.
Lynch, A.J., Mineral Crushing and Grinding Circuits. Elsevier, Amsterdam (1977) Leite, M.R.M., Moagem Nr~o Linear em Moinho de Barras. Ph. D. Thesis, University of Porto, Porto, Portugal (1984)
4.
.
Mika, T.S., Berlioz, L.M., & Fuerstenau, D.W., An Approach to the Kinetics of Dry Batch Ball Milling. 2nd European Symposium of Size Reduction. Verlag Chemie, Weinheim (1967)
.
Rocha E. Silva, J.A.C., Calibre: Conceito e Medida. Ph. D. Thesis, University of Porto, Porto, Portugal (1989)
.
Lynch, A.J., Simulation - The Design Tool for the Future. In Mineral Processing at a Crossroads (ed. B.A. Wills & R.W. Barley), Martinus Nijhoff, Dordrecht (1986)
.
Nash, J.C., Compact Numerical Methods for Computers. Adam Hilger, Ltd, Bristol (1979)
.
Leite, M.R.M, Circuitos de Fragmenta~o Grauda. Boletim de Minas, 21/2, Lisboa (1984)
180
M. R. MACHADO LEITE
10.
Whiten, W.J., The Simulation of Crushing Plants with Models Developed using Multiple Spline Regression. Journal of the South African Institute of Mining and Metallurgy, (May 1972)
11.
Greenwood, J., & Hiorns, F.J., A Comparison of Individual and Collective Breakage of Particle Assemblies. 2nd European Symposium of Size Reduction, Verlag Chemie, Weinheim (1967)
12.
Madureira, C.M.N., et al, Conceitos Fundamantais para o Estudo Fenomenol6gico dos Processos de Fragmenta~o. Modelos de Simula~o. II Simp. Sobre Teorias da Informa~o e dos Sistemas, Faculdade de Engenharia, University of Porto, Portugal (1972)
13.
Griffith, A.A., The Phenomena of Rupture and Flow in Solids. Phil Trans. A. (1921)
14.
Ferrara, G. et al, A Contribution to Screening Kinetics. XI International Mineral Processing Congress, University of Cagliari, Cagliari (1975)